## Abstract

It is known that the (Formula presented.) -sphere has at most (Formula presented.) combinatorially distinct triangulations with n vertices, for every (Formula presented.). Here we construct at least (Formula presented.) such triangulations, improving on the previous constructions which gave (Formula presented.) in the general case (Kalai) and (Formula presented.) for (Formula presented.) (Pfeifle–Ziegler). We also construct (Formula presented.) geodesic (a.k.a. star-convex) n-vertex triangulations of the (Formula presented.) -sphere. As a step for this (in the case (Formula presented.)) we construct n-vertex 4-polytopes containing (Formula presented.) facets that are not simplices, or with (Formula presented.) edges of degree three.

Original language | American English |
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Pages (from-to) | 737-762 |

Number of pages | 26 |

Journal | Mathematische Annalen |

Volume | 364 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Apr 2016 |

### Bibliographical note

Funding Information:Research of Nevo and Wilson was partially supported by Marie Curie grant IRG-270923 and ISF grant 805/11. Research of Santos was supported by the Spanish Ministry of Science (MICINN) through grant MTM2011-22792, and by a Humboldt Research Award of the Alexander von Humboldt Foundation.

Publisher Copyright:

© 2015, Springer-Verlag Berlin Heidelberg.